A Spherical Harmonic Discontinuous Galerkin Method for Radiative Transfer Equations with Vacuum Boundary Conditions

In this paper, we propose and analyze a spherical harmonic discontinuous Galerkin (SH-DG) method for solving the radiative transfer equations with vacuum boundary conditions. To incorporate vacuum boundary conditions in spherical harmonic approximations, we first embed the original domain into a larger computational area of rectangular type with an extra pure absorbing layer and then establish a perturbation problem with a periodic condition at the boundary of the extended domain. Since the outflow radiative intensity at the outer boundary of the extended area can be made arbitrarily small by sufficiently increasing the magnitude of absorption in or the thickness of the absorbing layer, such a replacement of the boundary condition only causes a minimal difference between the solution of the perturbation problem and the original problem in the original domain, but will benefit the construction of the discretization scheme. Then based on the analysis of the perturbation problem and the SH-DG method for solving the radiative transfer equation with periodic boundary conditions, the well-posedness and the error estimates are derived for the approximation solution arising from the SH-DG method for solving the radiative transfer equation with vacuum boundary conditions. Numerical examples with both periodic and vacuum boundary conditions are included to validate the theoretical results.